Theorem ipasslem1 24250

Description: Lemma for ipassi 24260. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	|- X = ( BaseSet ` U )
ip1i.2	|- G = ( +v ` U )
ip1i.4	|- S = ( .sOLD ` U )
ip1i.7	|- P = ( .iOLD ` U )
ip1i.9	|- U e. CPreHil OLD
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem1	|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Proof of Theorem ipasslem1

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 10608	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
2		ax-1cn 9359	. . . . . . . . . . . 12 |- 1 e. CC
3		ip1i.9	. . . . . . . . . . . . . 14 |- U e. CPreHil OLD
4	3	phnvi 24235	. . . . . . . . . . . . 13 |- U e. NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 |- X = ( BaseSet ` U )
6		ip1i.2	. . . . . . . . . . . . . 14 |- G = ( +v ` U )
7		ip1i.4	. . . . . . . . . . . . . 14 |- S = ( .sOLD ` U )
8	5, 6, 7	nvdir 24030	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( k e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
9	4, 8	mpan 670	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
10	2, 9	mp3an2 1302	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
11	1, 10	sylan 471	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
12	5, 7	nvsid 24026	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
13	4, 12	mpan 670	. . . . . . . . . . . 12 |- ( A e. X -> ( 1 S A ) = A )
14	13	adantl 466	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 S A ) = A )
15	14	oveq2d 6126	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k S A ) G ( 1 S A ) ) = ( ( k S A ) G A ) )
16	11, 15	eqtrd 2475	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G A ) )
17	16	oveq1d 6125	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) G A ) P B ) )
18		ipasslem1.b	. . . . . . . . . . . . 13 |- B e. X
19		ip1i.7	. . . . . . . . . . . . . 14 |- P = ( .iOLD ` U )
20	5, 19	dipcl 24129	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
21	4, 18, 20	mp3an13 1305	. . . . . . . . . . . 12 |- ( A e. X -> ( A P B ) e. CC )
22	21	mulid2d 9423	. . . . . . . . . . 11 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
23	22	adantl 466	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 x. ( A P B ) ) = ( A P B ) )
24	23	oveq2d 6126	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
25	5, 7	nvscl 24025	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ k e. CC /\ A e. X ) -> ( k S A ) e. X )
26	4, 25	mp3an1 1301	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( k S A ) e. X )
27	1, 26	sylan 471	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( k S A ) e. X )
28	5, 6, 7, 19, 3	ipdiri 24249	. . . . . . . . . . 11 |- ( ( ( k S A ) e. X /\ A e. X /\ B e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
29	18, 28	mp3an3 1303	. . . . . . . . . 10 |- ( ( ( k S A ) e. X /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
30	27, 29	sylancom 667	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
31	24, 30	eqtr4d 2478	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) G A ) P B ) )
32	17, 31	eqtr4d 2478	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) )
33		oveq1 6117	. . . . . . 7 |- ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
34	32, 33	sylan9eq 2495	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
35		adddir 9396	. . . . . . . . 9 |- ( ( k e. CC /\ 1 e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
36	2, 35	mp3an2 1302	. . . . . . . 8 |- ( ( k e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
37	1, 21, 36	syl2an 477	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
38	37	adantr 465	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
39	34, 38	eqtr4d 2478	. . . . 5 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) )
40	39	exp31 604	. . . 4 |- ( k e. NN0 -> ( A e. X -> ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
41	40	a2d 26	. . 3 |- ( k e. NN0 -> ( ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
42		eqid 2443	. . . . . 6 |- ( 0vec ` U ) = ( 0vec ` U )
43	5, 42, 19	dip0l 24135	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
44	4, 18, 43	mp2an 672	. . . 4 |- ( ( 0vec ` U ) P B ) = 0
45	5, 7, 42	nv0 24036	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( 0vec ` U ) )
46	4, 45	mpan 670	. . . . 5 |- ( A e. X -> ( 0 S A ) = ( 0vec ` U ) )
47	46	oveq1d 6125	. . . 4 |- ( A e. X -> ( ( 0 S A ) P B ) = ( ( 0vec ` U ) P B ) )
48	21	mul02d 9586	. . . 4 |- ( A e. X -> ( 0 x. ( A P B ) ) = 0 )
49	44, 47, 48	3eqtr4a 2501	. . 3 |- ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) )
50		oveq1 6117	. . . . . 6 |- ( j = 0 -> ( j S A ) = ( 0 S A ) )
51	50	oveq1d 6125	. . . . 5 |- ( j = 0 -> ( ( j S A ) P B ) = ( ( 0 S A ) P B ) )
52		oveq1 6117	. . . . 5 |- ( j = 0 -> ( j x. ( A P B ) ) = ( 0 x. ( A P B ) ) )
53	51, 52	eqeq12d 2457	. . . 4 |- ( j = 0 -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) )
54	53	imbi2d 316	. . 3 |- ( j = 0 -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) ) )
55		oveq1 6117	. . . . . 6 |- ( j = k -> ( j S A ) = ( k S A ) )
56	55	oveq1d 6125	. . . . 5 |- ( j = k -> ( ( j S A ) P B ) = ( ( k S A ) P B ) )
57		oveq1 6117	. . . . 5 |- ( j = k -> ( j x. ( A P B ) ) = ( k x. ( A P B ) ) )
58	56, 57	eqeq12d 2457	. . . 4 |- ( j = k -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) )
59	58	imbi2d 316	. . 3 |- ( j = k -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) ) )
60		oveq1 6117	. . . . . 6 |- ( j = ( k + 1 ) -> ( j S A ) = ( ( k + 1 ) S A ) )
61	60	oveq1d 6125	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j S A ) P B ) = ( ( ( k + 1 ) S A ) P B ) )
62		oveq1 6117	. . . . 5 |- ( j = ( k + 1 ) -> ( j x. ( A P B ) ) = ( ( k + 1 ) x. ( A P B ) ) )
63	61, 62	eqeq12d 2457	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) )
64	63	imbi2d 316	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
65		oveq1 6117	. . . . . 6 |- ( j = N -> ( j S A ) = ( N S A ) )
66	65	oveq1d 6125	. . . . 5 |- ( j = N -> ( ( j S A ) P B ) = ( ( N S A ) P B ) )
67		oveq1 6117	. . . . 5 |- ( j = N -> ( j x. ( A P B ) ) = ( N x. ( A P B ) ) )
68	66, 67	eqeq12d 2457	. . . 4 |- ( j = N -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
69	68	imbi2d 316	. . 3 |- ( j = N -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) ) )
70	41, 49, 54, 59, 64, 69	nn0indALT 10758	. 2 |- ( N e. NN0 -> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
71	70	imp 429	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   ` cfv 5437  (class class class)co 6110   CCcc 9299   0cc0 9301   1c1 9302   + caddc 9304   x. cmul 9306   NN0cn0 10598   NrmCVeccnv 23981   +vcpv 23982   BaseSetcba 23983   .sOLDcns 23984   0veccn0v 23985   .iOLDcdip 24114   CPreHil OLDccphlo 24231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fz 11457  df-fzo 11568  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-sum 13183  df-grpo 23697  df-gid 23698  df-ginv 23699  df-ablo 23788  df-vc 23943  df-nv 23989  df-va 23992  df-ba 23993  df-sm 23994  df-0v 23995  df-nmcv 23997  df-dip 24115  df-ph 24232

This theorem is referenced by:  ipasslem2  24251  ipasslem3  24252  ipasslem4  24253